Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123243 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
285 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,23,28)(19,29,32,47)(20,33,35,45)(21,25,22,27)(24,39,42,30)(26,37,44,36)(31,46,38,43)(34,40,48,41);(1,3,6,13)(2,7,10,15)(4,12,5,14)(8,11,9,16)(17,19)(18,24)(20,34)(21,31)(22,38)(23,32)(25,40)(26,43)(27,41)(28,42)(29,33)(30,36)(35,48)(37,39)(44,46)(45,47);(1,4)(2,8,10,9)(3,12)(5,6)(7,11,15,16)(13,14)(18,25,28,27)(20,35)(24,40,42,41)(26,33,44,45)(29,46,47,43)(30,39)(34,48)(36,37)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gi,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30469);